Majority Leader Mitch McConnell needs the support of 50 of 52 GOP senators to proceed to a floor debate on the bill, and two senators have already said they will not support that motion. That means just one more GOP senator coming out against the motion to proceed would stop the bill, as written, in its tracks.

Increasing the pressure: GOP leaders have restarted the clock by publicly stating that they'd like a vote (or at least to take the procedural steps toward a vote) next week. The White House is also making its push -- on Twitter, by phone and behind closed doors -- in an effort to net President Donald Trump a major legislative achievement ahead of lawmakers' August recess.

The White House legislative affairs team held meetings on the legislation with Senate leadership "all afternoon" on Friday, a senior White House official said. Trump and Vice President Mike Pence will be making calls throughout the weekend on health care as well, the official said.

McConnell recently announced that the recess would be delayed by at least two weeks, but it still wasn't clear going into the weekend whether the additional time would help the GOP leadership get this legislation through the chamber.

Here's what we know after this week:

The revised legislation also has $45 billion in opioid-treatment funding -- a top request from senators like Rob Portman of Ohio and Shelley Moore Capito of West Virginia -- and money for states meant to lower premiums for high-cost enrollees.

In several key areas, though, the new bill remains unchanged.

The new version does not address moderates' concerns about cuts to Medicaid, which the original bill would slash by $772 billion by 2026. That would leave an estimated 15 million fewer people insured by the program.

The Congressional Budget Office is expected to release its score of the revised bill early next week, when the White House will continue its lobbying effort, the senior White House official said.

Sen. Rand Paul of Kentucky: "The new bill is the same as the old bill -- except for worse," he said Wednesday. Paul, who reiterated his opposition Thursday, is facing pressure from the White House to support the bill. While on an official trip to Paris, Trump called Paul urging him to vote yes, an aid to Paul told CNN.

The revised Senate bill left Medicaid virtually untouched, meaning the serious concerns of moderates (particularly those in Medicaid-expansion states) were not addressed. And with the Cruz amendment in the bill, all eyes are now on moderates.

Heller is another one to watch. In the past, he has sharply criticized the Senate's version of the bill, and he's recently indicated that he's in lockstep with his state's Republican governor, Brian Sandoval. The state's chief executive this week did not sound enthused with the bill, particularly as it relates to residents in his state covered under the Obamacare-era expansion of Medicaid.

"I'm greatly concerned and very protective of the expansion population," Sandoval told CNN Thursday at the National Governors Association meeting in Rhode Island. "They are living healthier and happier lives as a result of their receiving coverage, and for them to lose that at this point would be very hurtful for them. And it's about people -- this is about people. And 210,000 people in my state."

In addition to these senators, a number of others have remained noncommittal. They include: Murkowski; Jeff Flake of Arizona; Mike Lee of Utah; John Hoeven of North Dakota; Bill Cassidy of Louisiana; Thom Tillis of North Carolina; Ben Sasse of Nebraska; Thad Cochran of Mississippi; Cory Gardner of Colorado; and Todd Young of Indiana.

The President tweeted four times about health care on Friday from France. He said Vice President Mike Pence is "working hard" on it and that he'll have his "pen in hand" to sign a bill into law.

Pence touted the Senate bill -- and stressed his view that the Affordable Care Act is collapsing -- during remarks Friday to a group of about 30 governors at their group's summer meeting.

"Let me be clear: President Trump and I believe the Senate health care bill strengthens and secures Medicaid for the neediest in our society, and this bill puts this vital American program on a path to long-term sustainability," he said, without noting that the bill also cuts Medicaid spending from current projections.

"I understand and appreciate, as the President does, the concerns that many of you have as we talk about Medicaid in the future going forward. Our administration's paid very close attention to this issue," he said.

The President remains focused on getting a deal on health care that's better for the American people, a senior administration adviser told CNN on Friday.

"If they don't get this done now, I don't know when it'll happen," the adviser said. "We've had seven years to create an alternative plan."

CNN's MJ Lee, Lauren Fox, Tami Luhby, Eric Bradner, Liz Landers, Ryan Nobles and Ted Barrett contributed to this report.

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Highlights: Crucial week for Republican health care plan ...